基于模糊约束的贝叶斯网络参数学习

doi:10.12305/j.issn.1001-506X.2023.02.15

Abstract

Abstract:

To address the problem of Bayesian networks parameter learning under small datasets, a fuzzy maximum posteriori estimation method is proposed, introducing fuzzy theory into parameter learning. The hyperparameter is determined by using the membership function to measure constraint effectiveness to improve the accuracy of constraint usage for learning. Experiments prove that the proposed method can effectively improve the accuracy of parameter learning. In addition, the proposed parameter learning method is applied to a network security assessment by using common vulnerability scoring system as expert priori parameters and combining vulnerability transfer samples to perform parameter learning. Finally, the node and path security evaluation verifies the effectiveness of the proposed algorithm.

Key words: Bayesian network, membership function, parameter learning, network security assessment

CLC Number:

TP181

Xinxin RU, Xiaoguang GAO, Yangyang WANG. Bayesian network parameter learning based on fuzzy constraints[J]. Systems Engineering and Electronics, 2023, 45(2): 444-452.

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References 32

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BNs	规模	节点	参数	约束
Earthquake	小	5	10	10
Cancer	小	5	10	10
Sachs	中	11	178	79
Boerlage92	中	23	86	73
Win95pts	大	76	574	222
Andes	大	223	1157	935

编号	约束
1	0.9≤P(V₁=低)≤1
2	P(V₂=真)＜P(V₂=假)
3	0.9≤P(V₃=假\|V₁=低, V₂=真)≤1
4	0.9≤P(V₃=假\|V₁=高, V₂=真)≤1
5	0.9≤P(V₃=假\|V₁=低, V₂=假)≤1
6	0.9≤P(V₃=假\|V₁=高, V₂=假)≤1
7	0.9≤P(V₄=阳性\|V₃=真)≤1
8	P(V₄=阳性\|V₃=假)＜P(V₄=阴性\|V₃=假)
9	P(V₅=真\|V₃=真)＞P(V₅=假\|V₃=真)
10	P(V₅=真\|V₃=假)＜P(V₅=假\|V₃=假)

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	1.173 3±0.766 6	0.348 1±0.057 5	0.044 2±0.022 7	0.105 4±0.047 8	0.080 5±0.032 8
Cancer	0.846 7±0.744 7	0.844 0±0.097 4	0.104 2±0.035 4	0.052 9±0.043 1	0.049 8±00.000 1
Sachs	0.983 2±0.235 8	0.392 1±0.024 4	0.352 6±0.099 5	0.198 3±0.027 8	0.147 3±0.010 3
Boerlage92	1.217 9±0.216 1	0.332 7±0.022 5	0.143 8±0.088 2	0.142 3±0.024 2	0.072 9±0.014 8
Win95pts	0.582 8±0.110 6	0.329 9±0.007 2	0.293 9±0.057 0	0.233 0±0.013 0	0.213 3±0.004 5
Andes	1.005 2±0.067 4	0.394 2±0.005 6	0.161 9±0.020 5	0.172 8±0.009 5	0.132 5±0.004 1
均值	0.968 2	0.440 2	0.183 4	0.150 8	0.116 1

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	0.936 7±0.898 3	0.406 6±0.065 0	0.014 8±0.011 4	0.121 5±0.049 2	0.060 0±0.019 7
Cancer	0.470 2±0.442 8	0.416 5±0.083 4	0.059 9±0.032 1	0.036 7±0.017 9	0.020 6±00.001 0
Sachs	0.603 3±0.155 3	0.430 9±0.029 8	0.183 4±0.080 0	0.126 2±0.019 4	0.089 8±0.008 4
Boerlage92	0.510 7±0.188 7	0.365 7±0.020 0	0.030 2±0.034 5	0.076 2±0.020 5	0.023 9±0.005 7
Win95pts	0.653 8±0.086 7	0.350 5±0.004 8	0.381 1±0.060 2	0.212 7±0.012 5	0.173 6±0.004 8
Andes	0.698 2±0.051 1	0.429 6±0.005 4	0.159 5±0.016 7	0.118 7±0.007 0	0.064 7±0.002 9
均值	0.645 5	0.416 7	0.138 2	0.115 3	0.072 1

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	0.377 4±0.364 4	0.431 5±0.058 2	0.014 2±0.016 2	0.083 4±0.028 4	0.056 6±0.014 1
Cancer	0.285 2±0.302 4	0.179 3±0.071 6	0.050 1±0.031 4	0.030 9±0.021 8	0.008 0±0.000 1
Sachs	0.437 3±0.106 2	0.477 8±0.019 8	0.136 1±0.038 4	0.093 7±0.013 4	0.065 4±0.007 4
Boerlage92	0.259 4±0.103 2	0.369 3±0.013 8	0.009 3±0.003 2	0.041 7±0.012 9	0.010 4±0.002 5
Win95pts	0.722 9±0.092 3	0.377 2±0.004 4	0.453 4±0.058 6	0.207 5±0.014 0	0.153 2±0.002 3
Andes	0.466 3±0.038 0	0.447 4±0.006 2	0.109 1±0.015 8	0.083 1±0.005 4	0.039 3±0.002 5
均值	0.424 8	0.380 5	0.128 7	0.090 1	0.055 5

Bayesian network parameter learning based on fuzzy constraints

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H	CVE	CVSS约束
H₂	CVE-2018-6036	P(H₂=1\|Π₁=1)=0.75
H₃	CVE-2017-14827	P(H₃=1\|Π₃=1)=0.68
H₄	CVE-2017-1336	P(H₄=1\|Π₄=1)=0.36
H₅	CVE-2017-11887	P(H₅=1\|Π₅=1)=0.26
H₆	CVE-2019-1010156	P(H₆=1\|Π₆=1)=0.64
H₇	CVE-2014-1761	P(H₇=1\|Π₇=1)=0.93
H₈	CVE-2018-16844	P(H₈=1\|Π₈=1)=0.78
H₉	CVE-2016-10627	P(H₉=1\|Π₉=1)=0.63
H₁₀	CVE-2016-3454	P(H₁₀=1\|Π₁₀=1)=0.76
H₁₁	CVE-2017-16375	P(H₁₁=1\|Π₁₁=1)=0.93
H₁₂	CVE-2018-12498	P(H₁₂=1\|Π₁₂=1)=0.75
H₁₃	CVE-2015-7994	P(H₁₃=1\|Π₁₃=1)=0.75
H₁₄	CVE-2019-1010246	P(H₁₄=1\|Π₁₄=1)=0.55
H₁₅	CVE-2017-8620	P(H₁₅=1\|Π₁₅=1)=0.93
H₁₆	CVE-2017-11873	P(H₁₆=1\|Π₁₆=1)=0.76

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BNs	CMLE	QMAP	FMAP
Earthquake	0.834 2± 0.563 1	0.117 0± 0.041 1	0.073 8± 0.041 7
Cancer	0.578 4± 0.355 7	0.075 3± 0.054 0	0.054 7± 0.040 5
Sachs	0.679 7± 0.138 5	0.194 3± 0.019 2	0.144 6± 0.008 4
Boerlage92	0.911 8± 0.216 6	0.153 2± 0.029 8	0.075 4± 0.011 8
Win95pts	0.352 2± 0.055 6	0.226 4± 0.009 7	0.214 7± 0.004 6
Andes	0.476 9± 0.045 5	0.190 3± 0.009 7	0.133 7± 0.003 1
均值	0.638 9	0.159 4	0.116 2

BNs	CMLE	QMAP	FMAP
Earthquake	0.494 7± 0.439 7	0.124 3± 0.059 8	0.065 3± 0.017 4
Cancer	0.373 5± 0.306 9	0.063 8± 0.028 4	0.014 7± 0.016 5
Sachs	0.394 0± 0.107 9	0.149 5± 0.021 0	0.095 0± 0.010 8
Boerlage92	0.292 1± 0.098 5	0.081 0± 0.018 3	0.031 2± 0.006 9
Win95pts	0.436 2± 0.049 3	0.210 6± 0.010 9	0.177 9± 0.004 1
Andes	0.293 2± 0.024 2	0.119 9± 0.008 5	0.068 0± 0.004 0
均值	0.380 6	0.124 9	0.075 4

BNs	CMLE	QMAP	FMAP
Earthquake	0.063 1± 0.095 5	0.109 0± 0.041 5	0.059 2± 0.019 6
Cancer	0.108 4± 0.145 4	0.047 6± 0.018 1	0.009 2± 0.006 5
Sachs	0.288 5± 0.046 6	0.110 3± 0.015 2	0.062 8± 0.009 4
Boerlage92	0.144 1± 0.062 8	0.050 7± 0.009 5	0.016 6± 0.003 7
Win95pts	0.501 0± 0.044 2	0.207 0± 0.010 6	0.154 8± 0.002 8
Andes	0.196 7± 0.023 3	0.100 3± 0.006 9	0.040 4± 0.002 8
均值	0.217 0	0.107 5	0.057 2

BNs	MLE	MAP	CMLE	QMAP	FMAP
Earthquake	0.001 7±0.000 2	0.001 7±0.000 3	1.494 4±0.024 5	0.002 1±0.000 1	0.002 8±0.000 5
Cancer	0.001 7±0.000 3	0.001 7±0.000 4	1.491 9±0.033 9	0.002 1±0.000 1	0.002 8±0.000 4
Sachs	0.015 5±0.000 7	0.015 5±0.000 7	3.840 7±0.048 5	0.016 4±0.001 2	0.037 2±0.002 5
Boerlage92	0.011 7±0.002 2	0.012 1±0.001 5	7.383 4±0.077 3	0.012 5±0.000 7	0.024 0±0.001 7
Win95pts	0.070 5±0.004 8	0.072 2±0.003 9	23.420 1±0.362 1	0.089 3±0.003 7	0.159 4±0.007 7
Andes	0.172 7±0.016 0	0.174 6±0.013 5	77.730 0±6.792 5	0.197 9±0.016 0	0.343 2±0.031 3
均值	0.045 8	0.046 3	19.226 8	0.053 38	0.094 9